These exercises are designed to introduce you to the concept of surface-to-volume
ratios (S/V) and their importance in biology. S/V ratio refers to the amount of surface a
structure has relative to its size. Or stated in a more gruesome manner, the amount of
"skin" compared to the amount of "guts". To calculate the S/V ratio,
simply divide the surface area by the volume.

EXERCISE 1. INFLUENCE OF SIZE ON S/V RATIOS. We will use a cube to serve as a
model cell (or organism). Cubes are especially nice because surface area (length x width x
number of sides) and volume (length x width x height) calculations are easy to perform. To
calculate the surface-to-volume ratio divide the surface area by the volume. Complete the
table below for a series of cubes of varying size:

Length of a side
(mm)

Surface Area (mm2)

Volume (mm3)

Surface/volume ratio

1

2

3

4

5

6

7

8

9

10

Questions and Analysis:

Which cube has the greatest surface area? volume? S/V ratio?

What happens to the surface area as the cubes get larger? What happens to the volume as
the cubes get larger? What happens to the S/V ratio as the cubes get larger?

Proportionately, which grows faster - surface area or volume? Explain.

Which cube has the most surface area in proportion to its volume?

If you cut a cube in half, how does the volume, surface area and S/V ratio of one of the
resultant halves compare to the original?

As the linear dimension of the cube triples, the surface area increase by the (square or
cube) of the linear dimension, and the volume increases by the (square or cube) of the
linear dimension.

EXERCISE 2. SHAPE AND S/V RATIOS: In this exercise we will explore the
impact of shape on surface to volume ratios. The three shapes given below have
approximately the same volume. For each, calculate the volume, surface area and S/V ratio
and complete the table.

If you had to select a package with the greatest volume and smallest surface area, what
shape would it be?

Explain why the shape of animals is basically "spherical", whereas plants and
fungi are "filamentous".

EXERCISE 3. S/V RATIOS IN FLATTENED OBJECTS: In this exercise we will explore
how flattening an object impacts the surface to volume ratio. Consider a box that is 8 x 8
x 8 mm on a side. Then, imagine that we can flatten the box making it thinner and thinner
while maintaining the original volume. What will happen to the surface area, and s/v ratio
as the box is flattened? Complete the table below.

Box No.

Height (mm)

Length (mm)

Width (mm)

Surface area (mm2)

volume (mm3)

s/v ratio

1

8

8

8

2

4

16

8

3

2

16

16

4

1

32

16

5

0.5

32

32

Questions/Analysis:

Explain why leaves are thin and flat.

Why do elephants have large, flat ears?

How do you think flatworms feed?

EXERCISE 4. S/V RATIOS IN ELONGATED OBJECTS: In this exercise we will explore
how elongating an object impacts the surface to volume ratio. Consider a box that is 8 x 8
x 8 mm on a side. Then, imagine that we pull on the ends to make it longer and longer
while maintaining the original volume. What will happen to the surface area, and s/v ratio
as the box is flattened? Complete the table below.

Box No.

Length (mm)

Height (mm)

Width (mm)

Surface area (mm2)

volume (mm3)

s/v ratio

1

8

8

8

2

16

4

8

3

32

4

4

4

64

2

4

5

128

2

2

Questions/Analysis:

Explain the shape of blood vessels.

Explain why roots have "hairs".

EXERCISE 5. WHY ARE CELLS SMALL? The typical eukaryotic cell is rather small -
approximately 100 mm in diameter. This exercise is designed to help provide an explanation
why cells are not normally larger.

Obtain 2 cell models, one small and one large. Measure the length and diameter of each
and then record your data in the table below. Place each cell in a bowl containing clear
vinegar (BE CAREFUL!). Allow to sit for a few minutes, or until most of the blue color is
gone from the smallest cell. Remove the models with a plastic spoon (CAUTION: don't get
the vinegar on your hands!!!!) and place it on a piece of paper towel. Then, measure the
size of the colored area remaining and record these data in the table below. Complete the
calculations.

Theory: The cell models are made of a gelatin-like material called agar. The agar has
an acid-sensitive dye incorporated into it. The dye turns from blue to yellow (clear in
the presence of acid). The uptake of acid, and hence colorless areas of the cell models
represents the uptake of food/nutrients by the cell. From this, we can calculate the
percent of each cell that was fed during the incubation period.

If a cell divides in half, how does the volume, surface area and S/V ratio of each new
cell compare to that of the original?

Explain why the rate of cell growth slows as a cell gets larger.

Explain why cells divide when they get large.

EXERCISE 6. WHY DO MICE HAVE GREATER METABOLIC RATES THAN ELEPHANTS? - It is
well known that there is an inverse relationship between body size and metabolic rate. The
purpose of this exercise is to determine the reason for this relationship.

Procedure:

Pack ice around the containers in the pan to a depth of about two inches. Then add cold
water to bring the water level up to the top of the ice.

Turn on the hot water tap and let it run until the temperature reaches 55-65 (measure
with your thermometer).

Measure 8 mL of hot water with a pipet and place it into the 30 mL flat-bottomed vial.

Place the thermometer in the hot water. At first the temperature will rise. When it
begins to fall again, measure the temperature. If it is below 40 C, start again with
hotter water.

When the temperature reaches 40 C, record the time. Exactly two minutes later record the
temperature again. Record your data in the table below.

Measure 80 mL of hot water with a graduate cylinder and transfer it to the larger (100
mL) beaker. Then repeat steps 4 & 5.

Calculate the initial and final heat content (rows 8 and 10) by multiplying the
temperature (rows 7 and 9) by the volume of water (row 4) by the specific heat of water
(4.2 joules/cm3 C). [Note - specific heat is an indication of the amount of
heat it takes to change the temperature of a substance.] Thus, this equation is:

Calculate the total heat loss during the two minute period (row 11) by subtracting the
final heat content (row 10) from the initial heat content (row 8).

Calculate the relative heat loss (joules lost/cm3) for each volume (row 13)
by dividing the total heat loss in column d by the volume of the vessel (row 4).

Lastly, measure the height (row 1) and diameter (row 2) of the water in each container
and then calculate radius (row 3), volume (Row 4), surface area (row 5) and S/V ratio (row
6) for both containers.

Small

Large

1. height (cm)

2. diameter (cm)

3. radius (cm)

4. volume (cm3)

8

80

5. surface area (cm2)

6. S/V ratio

7. initial temp (C)

40

40

8. initial heat content (J)

9. final temp (C)

10. final heat content (J)

11. total heat loss (Joules/2 minute)

12. total heat loss (Joules/min)

13. relative heat loss (kJ/min/cm3)

Analysis/Questions:

Which vessel had the greatest volume? surface area? S/V ratio?

Which vessel lost more total heat? Did you expect that? Explain.

Which vessel lost the most heat relative to its size? Did you expect that? Explain.

Which vessel showed the greatest temperature change?

To maintain a constant temperature in the vessels, which would have required the most
total heat input? Which would have required the most heat proportional to its size?

Who will loose more total heat in a given period, an infant or an adult? A mouse or an
elephant? Explain.

Who will loose more heat relative to its volume, an infant or an adult? A mouse or an
elephant? Explain.

Who will need to eat the most food, an infant or an adult? A mouse or an elephant?
Explain. Who will need to eat the most food relative to size? Explain.

Explain why small animals have a higher metabolic rate than large animals

Explain why shrews are voracious feeders.

ADDITIONAL QUESTIONS:

Explain why cats can fall off of tall buildings and survive. Why do people splat?

Explain the advantages/disadvantages of block vs cube ice.

Describe the scientific inaccuracy in the episode of Goldilocks and the porridge.

Explain why lungs, gills and intestines have the shape they do.

Explain why plants are essentially a cluster of filaments, whereas animals are blobs. In
other words, why is a thin, elongated rectangle a good model for a plant, but a square a
good model for an animal?

Describe the shape of a radiator? Explain why it has this shape.

Medieval churches were often built in the shape of a crucifix. Explain why.

Mice have large eyes relative to size and elephants small ones. Explain why. Are large
eyes better than small ones?

The earth is geologically active (has a molten core) but the moon is apparently no
longer geologically active. Explain why using S/V ratios.

Based on size relationships, explain why a human is smarter than a mouse.

Shrews have a reputation for being "mean". In other words, they must feed
constantly. Explain why.

Why are there few small animals in the arctic?

Why would a mouse-sized human lack intelligence?

Explain how S/V ratios relate to the form of plants that have evolved in mesic, xeric
and hydric environments.